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2.2. Related work on consensus functions Theoretical approach Consensus functions VMA (Dimitriadou, Weingessel, and Hornik, 2002) BagClust1 (Dudoit and Fridlyand, 2003) Voting URCV, RCV, ACV (Ayad and Kamel, 2008) Also in (Fischer and Buhmann, 2003; Greene and Cunningham, 2006) CSPA, HGPA, MCLA (Strehl and Ghosh, 2002) Graph partitioning HBGF (Fern and Brodley, 2004) BALLS (Gionis, Mannila, and Tsaparas, 2007) EAC (Fred and Jain, 2005) CSPA (Strehl and Ghosh, 2002) BagClust2 (Dudoit and Fridlyand, 2003) Object co-association IPC (Nguyen and Caruana, 2007) BALLS (Gionis, Mannila, and Tsaparas, 2007) Majority Rule, CC Pivot (Goder and Filkov, 2008) Also in (Greene et al., 2004) QMI (Topchy, Jain, and Punch, 2003) Categorical clustering ITK (Punera and Ghosh, 2007) EM (Topchy, Jain, and Punch, 2004) PLA (<strong>La</strong>nge and Buhmann, 2005) Probabilistic Also in (Long, Zhang and Yu, 2005; Li, Ding and Jordan, 2007) Reinforcement learning (Agogino and Tumer, 2006) ALSAD, KMSAD, SLSAD Similarity as data (Kuncheva, Hadjitodorov, and Todorova, 2006) IVC, IPVC, IPC (Nguyen and Caruana, 2007) Centroid based Also in (Hore, Hall, and Goldgof, 2006) Agglomerative, Furthest, LocalSearch Correlation clustering (Gionis, Mannila, and Tsaparas, 2007) Search techniques SA, BOEM (Goder and Filkov, 2008) BestClustering (Gionis, Mannila, and Tsaparas, 2007) Cluster ensemble component selection BOK (Goder and Filkov, 2008) Also in (Fern and Lin, 2008) Table 2.1: Taxonomy of consensus functions according to their theoretical basis. times the limits between them are somewhat vague. Throughout the following paragraphs, the main features of these consensus functions are described. 2.2.1 Consensus functions based on voting The main idea underlying consensus functions based on voting strategies is the notion that objects assigned to a particular cluster by many partitions in the ensemble should also be located in that cluster according to the consensus clustering solution. One obvious way to achieve this goal is to consider cluster labels as votes, thus consolidating different clusterings by means of voting procedures. However, due to the symbolic nature of clusters (caused by the unsupervised nature of the clustering problem), it is necessary to disambiguate the clusters across the l components of the cluster ensemble prior to voting. One of the pioneering works in voting-based consensus clustering was the voting-merging algorithm (VMA) of Dimitriadou, Weingessel, and Hornik (2001). In that work, cluster 34
Chapter 2. Cluster ensembles and consensus clustering desambiguation is conducted by matching those clusters sharing the highest percentage of objects, iterating this cluster matching process across all the partitions in the cluster ensemble. As a result of the voting step, a fuzzy partition of the data set is obtained. Subsequently, a merging procedure is conducted on this soft partition, fusing those clusters which are closest to each other. By imposing a stopping criterion based on the sureness of the obtained clusters, this merging process is capable of finding the natural number of clusters in the data set. In (Dimitriadou, Weingessel, and Hornik, 2002), the authors define the consensus clustering solution as the one minimizing the average square distance with respect to all the partitions in the cluster ensemble. They demostrate that obtaining such consensus clustering boils down to finding the optimal re-labeling of the clusters of all the individual clusterings, which becomes an unfeasible problem if approached directly, since it requires an enumeration of all possible permutations. Therefore, they resort to the VMA consensus function of (Dimitriadou, Weingessel, and Hornik, 2001) for finding an approximate solution to the problem, extending its application to soft cluster ensembles. One of the two consensus functions proposed in (Dudoit and Fridlyand, 2003), called BagClust1, is based on applying plurality voting on the labelings in the cluster ensemble after a label disambiguation process based on measuring the overlap between clusters. The generation of the cluster ensemble components follows a resampling strategy similar to bagging, aiming to reduce the variability in the partitioning results via consensus clustering. A related proposal was the one presented in (Fischer and Buhmann, 2003). In that work, consensus clustering is viewed as a means for improving the quality and reliability of the results of path-based clustering, applying bagging for creating the hard cluster ensemble. The consensus clustering solution is obtained through a maximum likelihood mapping in which the label permutation problem is solved by means of the Hungarian method (Kuhn, 1955), which somehow resembles the application of plurality voting on the disambiguated individual partitions in the cluster ensemble (Ayad and Kamel, 2008). Moreover, a related reliability measure chooses the number of clusters with the highest stability as the preferable consensus solution. In (Greene and Cunningham, 2006), a majoritary voting strategy was applied for generating the consensus clustering solution, after disambiguating the clusters using the Hungarian algorithm. An additional interest of that work is that it was one of the first research efforts that considered the problem of creating and combining a large number of clustering solutions in the context of high dimensional data sets (such as text document collections). Indeed, the authors point out that using large ensembles boosts computational cost, while small ensembles tend to produce unstable consensus clustering solutions. In this context, the authors propose basing the cluster ensemble construction and the consensus clustering tasks on a prototype reduction technique that allows representing the whole data set by means of a minimal set of objects, ensuring that the clustering results will approximate those that would be obtained on the original data set. By doing so, the final clustering solution can be extended to those objects that have been left out of the reduced data set representation while alleviating the overall computational cost of the whole process. In particular, the reduced version of the cluster ensemble is obtained by projecting the pairwise object similarity matrix by means of a kernel matrix. The recent work of (Ayad and Kamel, 2008) presented a set of consensus functions based on cumulative voting –named URCV, RCV and ACV– whose time complexity scales linearly with the size of the data set. Another interesting feature is their capability of combining 35
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C.I.F. G: 59069740 Universitat Ramo
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Resum En segmentar de forma no supe
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Abstract When facing the task of pa
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Contents 2.2.6 Consensus functions
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Contents A.5 Consensus functions .
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Contents D.2.5 WDBC data set . . .
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List of Tables 3.13 Relative percen
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3.3. Deterministic hierarchical con
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3.3. Deterministic hierarchical con
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3.5. Discussion Consensus Consensus
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3.5. Discussion separate clustering
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Chapter 4 Self-refining consensus a
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Chapter 4. Self-refining consensus
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- What do we want to measure? Chapt
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φ (NMI) φ (NMI) φ (NMI) φ (NMI)
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φ (NMI) 0.8 0.78 0.76 0.74 0.72 0
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1. Given a cluster ensemble E conta
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%of experiments relative % φ (NMI)
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Publisher: Springer Series: Lecture
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5.1. Generation of multimodal clust
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5.2. Self-refining multimodal conse
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5.3. Multimodal consensus clusterin
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5.4. Discussion Data set Relative
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5.5. Related publications modes joi
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Chapter 6. Voting based consensus f
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6.2. Adapting consensus functions t
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6.3. Voting based consensus functio
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6.4. Experiments λc = 1 1 1 3 3 3
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6.4. Experiments Data set Soft clus
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6.4. Experiments CSPA EAC HGPA MCLA
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6.5. Discussion solutions on hard c
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Chapter 7 Conclusions The contribut
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Chapter 7. Conclusions of-the-art c
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Chapter 7. Conclusions Though put f
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Chapter 7. Conclusions clustering r
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Chapter 7. Conclusions hardened. Ou
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References Ben-Hur, A., D. Horn, H.
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References Deerwester, S., S.-T. Du
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References Fred, A. and A.K. Jain.
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References Ingaramo, D., D. Pinto,
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References Li, S.Z. and G. GuoDong.
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References Sebastiani, F. 2002. Mac
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References Topchy, A., A.K. Jain, a
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Appendix A Experimental setup A.1 T
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Appendix A. Experimental setup g. s
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A.2.1 Unimodal data sets Appendix A
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A.2.2 Multimodal data sets Appendix
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Appendix A. Experimental setup gene
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Appendix A. Experimental setup orig
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Appendix A. Experimental setup Data
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B.1. Clustering indeterminacies in
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C.1. Configuration of a random hier
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D.1. Experiments on consensus-based
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D.2. Experiments on selection-based
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E.1. CAL500 data set φ (NMI) 1 0.8
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E.1. CAL500 data set φ (NMI) CSPA
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E.2. InternetAds data set φ (NMI)
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E.3. Corel data set φ (NMI) 1 0.8
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F.1. Iris data set φ (NMI) 1 0.8 0
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F.3. Glass data set CSPA EAC HGPA M
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F.5. WDBC data set φ (NMI) 1 0.8 0
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F.7. MFeat data set φ (NMI) 1 0.8
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F.9. Segmentation data set φ (NMI)
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F.10. BBC data set φ (NMI) 1 0.8 0
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F.11. PenDigits data set φ (NMI) 1
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